Forbidden Patterns in Mixed Linear Layouts

TL;DR

This work addresses forbidden-pattern characterizations for mixed linear layouts of ordered graphs by introducing thick patterns as a generalization of twists and rainbows. It proves that for graphs with bounded maximum degree Δ, the mixed page number is bounded if and only if the largest thick pattern is bounded, with quantitative bounds such as mn(G) ∈ [k, O(Δ k^{8k-7} log^k k)]. It also establishes that a finite obstruction set characterization with the identity binding function cannot hold in general (for k ≥ 2, the obstruction family is infinite), while finite obstruction sets exist in the separated-layout setting for certain parameters. The paper develops quotient techniques and diamond/thick-pattern analyses to relate mixed layouts to poset decompositions and grid representations, providing constructive bounds and highlighting connections to data-structure processing and circle-graph coloring. These results advance understanding of mixed layouts and pave the way for further work on unbounded-degree graphs and upward planar applications.

Abstract

An ordered graph is a graph with a total order over its vertices. A linear layout of an ordered graph is a partition of the edges into sets of either non-crossing edges, called stacks, or non-nesting edges, called queues. The stack (queue) number of an ordered graph is the minimum number of required stacks (queues). Mixed linear layouts combine these layouts by allowing each set of edges to form either a stack or a queue. The minimum number of stacks plus queues is called the mixed page number. It is well known that ordered graphs with small stack number are characterized, up to a function, by the absence of large twists (that is, pairwise crossing edges). Similarly, ordered graphs with small queue number are characterized by the absence of large rainbows (that is, pairwise nesting edges). However, no such characterization via forbidden patterns is known for mixed linear layouts. We address this gap by introducing patterns similar to twists and rainbows, which we call thick patterns; such patterns allow a characterization, again up to a function, of mixed linear layouts of bounded-degree graphs. That is, we show that a family of ordered graphs with bounded maximum degree has bounded mixed page number if and only if the size of the largest thick pattern is bounded. In addition, we investigate an exact characterization of ordered graphs whose mixed page number equals a fixed integer $ k $ via a finite set of forbidden patterns. We show that for every $ k \ge 2 $, there is no such characterization, which supports the nature of our first result.

Figures (10)

• Figure 1: The two 3-thick patterns: three pairwise crossing 3-rainbows (left) and three pairwise nesting 3-twists (right)
• Figure 2: 2-$\diamondtimes$-patterns for \ref{['thm:fixed_char']}
• Figure 3: Left: A grid representation of a matching $M$ with edges indicating the comparabilities in the poset $P(M)$ of $M$. The 2-$\diamondtimes$-pattern is highlighted red. Right: The Ferrer's diagram of $M$ showing the partition $|P(M)| = 5 + 3 + 1$ by rows of length 5, 3 and 1. The diagram expresses that one chain can cover five elements (bottommost row), two chains can cover $5 + 3 = 8$ chains (two bottommost rows), and that three chains can cover all $5 + 3 + 1 = 9$ elements (all three rows). Note, however, that eight elements cannot be covered with two chains of length 5 and 3. The largest square is of size $2 \times 2$ and can be filled with elements of $P(M)$ representing a $2$-$\diamondtimes$-pattern by \ref{['lem:square_diamond']}. $C$ and $A$ denote the a maximum set of elements that can be covered by two chains, respectively two antichains, and the sizes of $C \setminus A$ and $A \setminus C$ are highlighted.
• Figure 4: Left: A grid representation of a matching, where some (in)comparabilities in the corresponding poset are shown with solid (dashed) edges. Right: The corresponding Ferrer's diagram. The matching contains a $\diamondtimes$-pattern of size $3 \times 3$ but its Ferrer's diagram contains only a $2 \times 2$-square. In this case, \ref{['lem:ferrer_ub']} gives a stronger upper bound of $2 \cdot 2$ instead of $2 \cdot 3$ as guaranteed by \ref{['thm:fixed_char']}.
• Figure 5: Construction of a separated matching in grid presentation with mixed page number $2k$ that contains no $\diamondtimes$-pattern of size larger than $k \times k$. Left: high-level idea of the construction: given a $k$-$\diamondtimes$-pattern (black) extend the chains to the top right (green) and the antichains to the top left (blue). Right: concrete construction for $k = 3$

Theorems & Definitions (31)

• Theorem 1: Gya85Dav22
• Theorem 2: HR92
• Theorem 3
• Theorem 4
• Theorem 5: name=\ref{['sec:diamond']}
• Theorem 6: name=\ref{['sec:thick']}
• Theorem 7: name=\ref{['sec:general_matching']}
• Theorem 8: name=\ref{['sec:general_matching']}
• Theorem 9: name=\ref{['sec:critical_separated']}
• Theorem 10: name=\ref{['sec:critical_general']}